SeriesNumber
43
Problem 1
Taska’s
new rice-cakes are so popular that the company can not keep up with the demand.
Regional demands shown in the table below total 1690 boxes per week, but Taska
can only supply 64% of that number.
Coventry
Birmingham
Nottingham
Leicester
Demand
410
330
640
310
Profit
(£)
1.20
1.50
1.70
1.50
The
table also shows different profit levels per box of rice-cakes for each region.
Taska wants to fulfill between 59% and 75% of each region’s demand.
What
is the maximal weekly profit that the company can achieve?
Problem 2
A
local appliance store has decided on an advertising campaign utilising
newspaper and radio. Each pound spent on newspaper advertising is expected to
reach 50 people in ‘Under £25,000’ and 20 in the ‘Over £25,000’ bracket.
Each
pound spent on radio advertising is expected to reach 30 people in ‘Under
£25,000’ and 40 in the ‘Over £25,000’ bracket.
If
the store wants to reach at least 125000 people in the ‘Under £25,000’ and at
least 95000 in the ‘Over £25,000’ bracket, what is the minimum cost of the
advertising campaign?
Problem 3
A
trust officer for a bank in West Midlands wants to invest in the following
bonds:
Bond
A
B
C
D
E
Yield
7%
10%
11%
6%
10%
Maturity
Short
Short
Long
Short
Long
Risk
High
High
High
High
Low
Tax-Free
No
Yes
No
No
Yes
She
has £110000 available for the investments. To achieve a diversity in her
investments she decided to satisfy the following constraints:
– At
least £24000 must be placed in short maturity bonds.
– No
more than £34000 may be invested in high-risk bonds.
–
Total funds invested in
low-risk bonds must be less than or equal to total funds placed in
long-maturity bonds.
The interest income derived
from tax-free bonds must be at least 20% of the total income. What is the
maximal return (in £) that the bank can achieve?
Problem 4
The Archer family raises cattle on their
farm in West Midlands. They also have a large garden in which they grow
ingredients for making two types of relish – SauceA and SauceB. These they sell
at local stores.
The
profit per kilogram of SauceA is £2.5 and the profit per kilogram of SauceB is
£4.25. The ingredients in each relish are cabbage, tomatoes, onions, and oil.
One kilogram of SauceA must contain at least 60% but no more than 70% cabbage,
and at least 7% onion, and at least 9% oil.
One kilogram of SauceB must contain at
least 60% but no more than 75% tomatoes, and at least 5% onion, and at least 5%
oil. Both relishes contain no more than 11% onion and no more than 12% oil.
The
family has enough time to make no more than 640 kilograms of relish. They know
also that they will sell at least 30% more SauceA than SauceB. They will have
this year 400 kilograms of cabbage, 380 kilograms of tomatoes, and 200
kilograms of onion. They can use any amount of oil needed.
What
is the maximal profit that the family can gain by producing and selling the
relish?
Problem 5
A mining company owns two mines, both of
which produce three grades of ore – high, medium, and low.
The company has a contract to supply a
smelting company with at least 80 tonnes of high-grade ore, 20 tonnes of
medium-grade ore, and 55 tonnes of low-grade ore.
Each
mine produces a certain amount of each type of ore each hour it is in
operation.
Mine
1 produces 7 tonnes of high-grade, 25 tonnes of medium-grade, and 15 tonnes of
low-grade ore per hour.
Mine
2 produces 25 tonnes of high-grade, 4 tonnes of medium-grade, and 15 tonnes of
low-grade ore per hour.
It
costs £200 per hour to mine each tonne of ore from mine 1, and it costs £160
per hour to mine a tonne of ore from mine 2 (i.e. one hour of work of mine 1
costs 200*(7+25+15)).
What
is the minimum company’s cost at which the contractual obligations can be met?
Problem 6
The
marketing manager of a North America Brewery must determine how many television
spots and magazine ads to purchase within an advertising budget of $120000.
Each
spot is expected to increase sales by 40000 cans, whereas each magazine ad will
account for 90000 cans in sell. The company’s gross profit on sales is $0.15
per can.
One
television spot costs $1800; each
magazine ad requires an expenditure of $1400.
To
have a balanced marketing program, the advertising budget must involve no more
than $52000 in magazine ads and no more than $52000 in television spots. What
is the maximal net profit (gross profit minus ads expenses) that the company
can achieve making the right decision?
(Note:
Number of units of ads bought can be fractional.)
Problem 7
A
production company has contracted to supply quantities of product A and product
B to local shops, together amounting to 8000 units. The market reseach shows
that the ratio of product A to product B can not exceed 3:1.
The company’s present
one-shift capacity is not large enough to produce the requested units,
therefore the company must use overtime and possibly subcontracting with othe
production companies. The result is an increase in the production cost as shown
in the following table.
Product
Production
type
Weekly
production range (units)
Unit
cost
Regular
<=600 £ 1.60 A Overtime ...<=750 £ 2.50 Subcontructing ...<=? £ 3.50 Regular <=800 £ 1.70 B Overtime ...<=1550 £ 2.50 Subcontructing ...<=? £ 3.70 The information in the table can be interpreted as follows. If the company produces, for example, 650 units of product A, then the production cost for first 600 units is £1.6 per unit and the production cost for the 50 units left is £2.5 per unit. What is the minimal cost that the company has to pay, if it designs an optimal production plan? Problem 8 A manufacturing company has contract to supply a customer with parts from May through September. However, the company does not have enough storage space to store the parts during this period, so it needs to rent extra warehouse space during the five-month period. Following are the company's additional space requirements. Month Space (sq.m.) May 47000 June 47000 July 30000 August 50000 September 40000 The rental agent the company is dealing with has provided it with the following cost schedule for warehouse space. This schedule shows that the longer the space is rented the cheaper the monthly priceis. Rental Period (months) £/sq.m./month 5 £0.75 4 £0.95 3 £1.10 2 £1.20 1 £1.35 The company can rent any amount of warehouse space on a monthly basis at any time for any number of whole months. For example, the company can rent in May 10000 square meters for one month and 37000 square meters for three months. What is the minimal total cost the company has to pay to meet the space requirements above? Problem 9 An investment firm has £200000 to invest in stocks, bonds, certificates of deposit, and real estate. The firm wishes to determine the mix of investments that will maximise the cash value at the and of six years. Opportunities to invest in stocks and bonds will be available at the beginning of each of the next six years. Each pound invested in stocks will return £1.2 ( a profit of £0.2) two years later; the return can be immediately reinvested in any alternative. Each pound invested in bonds will return £1.35 three years later; the return can be immediately reinvested in any alternative. Opportunities to invest in certificates of deposit will be available only once, at the beginning of the second year. Each pound invested in certificates will return £2 four yeas later, the return can be immediately reinvested in any alternative. Opportunities to invest in real estate will be available at the beginning of forth and sixth years. Each pound invested will return £1.35 one year later; the return can be immediately reinvested in any alternative. To minimise risk, the firm has decided to diversify its investments. The total amount invested in stocks can not exceed £100000, the total amount invested in real estate should be at least £20000, and the total amount invested in bonds should be at least £40000. What is the maximal cash value (initial sum plus profit) that the firm can get at the end of the six years period (i.e. at the beginning of seventh year)? Problem 10 Sunco Oil produces oil at two wells. Well 1 can produce as many as 183000 barrels per day, and well 2 can produce as many as 186000 barrels per day. It is possible to ship oil directly from the wells to Sunco's customers in Los Angeles and New York. Alternatively, Sunco could transport oil either to the port of Mobile or Galveston and then ship it by tanker to New York or Los Angeles. Transhipment capacities through Mobile are bounded by 70500 barrels per day; capacities in Galveston are unlimited. Los Angeles requires 141000 barrels per day, and New York requires 134000 barrels per day. The costs of shipping 1000 barrels between two points are shown in the table below. To From Mobile Galveston N.Y. L.A. Well 1 9 13 35 32 Well 2 30 24 20 22 Mobile 14 7 Galveston 17 13 What is the minimal transportation costs of meeting oil demands in Los Angeles and New York?
